Trace and norm for reduced elements of Jordan pairs

Let V be a Jordan pair over a field. An element x is called reduced if the principal inner ideal determined by x equals the line spanned by x. The linear span R = (R+, R-) of the reduced elements is an ideal of V, called the reduced part. A reduced x in V+ determines a linear functional fx on V- by
Qxv = fx(v)x.
K. McCrimmon showed that there is a natural bilinear form f on R+ x R-, the Faulkner form, with the property that f(x,y) = fx(y) for all reduced x,y. It is natural to ask whether there is in fact a bilinear form F on R+ x V- satisfying F(x,v) = fx(v) for all v in V- and reduced x in V+, and hence extending the Faulkner form. We give a positive answer to this question.

By interchanging the roles of V+ and V- we obtain an analogous bilinear form Fop on R- x V+ and then define the trace form Tr(u,v) if either u or v belongs to the reduced part by

Tr(u,v) = F(u,v) or Tr(u,v) = Fop(v,u),
respectively. The well-known properties of the Faulkner form carry over to the trace form. We also show that the elements of R are properly algebraic and compute Tr(u,v) in terms of the minimum polynomial of (u,v).

If the base field has characteristic 0, the exponential trace formula can be used to define a norm function N on R+ x V-. If V is nondegenerate this has the properties of the generic norm of a finite-dimensional Jordan pair: It is a polynomial in a generalized sense, serves as a denominator for the quasi-inverse, and satisfies well-known multiplicative identities.

An interesting application of the present theory is to complex Banach Jordan pairs where the reduced part is precisely the socle. In particular, if V = (J,J) is the Jordan pair of a unital nondegenerate Banach Jordan algebra J then tr(a) = Tr(a,1) and det(1-a) = N(a,1) are the trace and determinant functions introduced by B. Aupetit in by analytic methods.

(This paper has appeared in Comm. Algebra 25 (1997), 3011-3042, reprints are obtainable from the author)

Ottmar Loos < ottmar.loos@uibk.ac.at >